In order to test some implementations of numerical solvers for advection-diffusion equations with non-constant coefficients, I'm looking for examples of equations+border and initial conditions of this type which have explicit solutions. Could you propose any or references to such?
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The method of manufactured solution is very simple. If you have your system $F(u)=0$, then you let your solution be e.g. $u=\sin(x^2)+\cos(\exp(y))$, plug into $F$ to get $F(u)=g(x,y)$. Now $F_0=F(u)-g=0$ is your new equation with solution $u$ and corresponding BC.
I can suggest you the following studies where you can find the exact solution of the numerical examples.
1) Vit Dolejsi, hp-DGFEM for nonlinear convection-diffusion problems, http://dx.doi.org/10.1016/j.matcom.2013.03.001
2) Melanie Bittl, Dmitri Kuzmin, Roland Becker, The CG1-DG2 method for convection–diffusion equations in 2D, http://dx.doi.org/10.1016/j.cam.2014.03.008
3) Tie Zhang, Ying Sheng, Superconvergence and gradient recovery for a finite volume element method for solving convection-diffusion equations, DOI: 10.1002/num.21862
I hope they will work...